Calabi introduced the notion of an extremal metric, as an optimum, canonical, Kahler metric in a given Kahler class on a compact complex manifold. Well-known conjectures relate the existence of such a metric to  "stability" conditions of  the complex structure. In the case of toric manifolds these questions----involving differential geometry, PDE and algebraic geometry---can be formulated in an elementary way, in terms of convex geometry on the corresponding polytope. Moreover,  very recent work of Chen and Cheng leads to a general existence theorem in this setting. In the lecture we will survey these developments and explain some of the further questions that arise.